Mathematical and numerical modelling of rapid transients at partially lifted sluice gates

Abstract

In the present paper, we study the mathematical and numerical modelling of rapid transients at partially lifted sluice gates in the context of one-dimensional Shallow water Equations. First, improved exact solutions of the dam-break problem with partially lifted gate are presented, assuming (i) the dependence of the gate contraction coefficient on the upstream flow depth, and (ii) a viable definition of the submerged flow equation. We show that an exact dam-break solution always exists for any set of initial conditions, but there are also initial conditions for which the solution is multiple. To cope with this case, a novel disambiguation criterion based on the continuous dependence of the solution on the initial conditions is used to select the physically congruent choice amongst the alternatives. Finally, we present a one-dimensional Finite Volume numerical model for the approximate solution of the Shallow water Equations equipped with an approximate Riemann solver for the sluice gate treatment at cells interfaces. The numerical experiments show that the embedding of the steady state gate equations (equilibrium approach) into the approximate Riemann solver leads to unsatisfactory dam-break simulations, while a novel relaxed version of the gate equations (non-equilibrium approach) supplies results that are in agreement with both the novel exact solutions and the laboratory dam-break results available from the literature.